Heuristic Algorithms for Solving the Resource-Constrained Project Scheduling Problem: Classification and Computational Analysis

The resource constrained project scheduling problem (RCPSP) can be given as follows. A single project consists of a set J = {0,1, . . . ,n,n+1} of activities which have to be processed. Fictitious activities 0 and n+1 correspond to the “project start” and to the “project end”, respectively. The activities are interrelated by two kinds of constraints. First, precedence constraints force activity j not to be started before all its immediate predecessor activities comprised in the set P j have been finished. Second, performing the activities requires resources with limited capacities. We have K resource types, given by the set K = {1, . . . ,K}. While being processed, activity j requires r j,k units of resource type k ∈ K during every period of its non–preemptable duration p j. Resource type k has a limited capacity of Rk at any point in time. The parameters p j, r j,k, and Rk are assumed to be deterministic; for the project start and end activities we have p j = 0 and r j,k = 0 for all k ∈ K . The objective of the RCPSP is to find precedence and resource feasible completion times for all activities such that the makespan of the project is minimized. Figure 1 gives an example of a project comprising n = 6 activities which have to be scheduled subject to K = 1 renewable resource type with a capacity of 4 units. A feasible schedule with an optimal makespan of 13 periods is represented in Figure 2. Let Fj denote the finish time of activity j. A schedule S is given by a vector of finish times (F1,F2, . . . ,Fn). Let A(t) = { j ∈ J | Fj− p j ≤ t < Fj } be the set of activities which are being processed (active) at time instant t. We now can provide the conceptual decision model (1) – (4) (cf. Christofides et al. 1987).